Standard step method: Difference between revisions

Revision as of 14:40, 9 May 2013

Revision as of 14:40, 9 May 2013

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{{AFC submission|h||ts=20130425000651|u=NateJonesAR|ns=5}}

'''The Standard Step Method''' (STM) is a computational technique utilized to estimate one dimensional surface water profiles in open channels with gradually varied flow under steady state conditions. It uses a combination of the energy, momentum, and continuity equations to determine water depth with a given a friction slope <math>(S_f)</math>, channel slope <math>(S_0)</math>, channel geometry, and also a given flow rate. In practice, this technique is widely used through the computer program [[HEC-RAS]], developed by the US Army Corps of Engineers Hydrologic Engineering Center (HEC). <ref></ref><ref>{{cite web|last=USACE|title=HEC-RAS Version 4.1 User's Manual|publisher=Hydrologic Engineering Center, Davis, CA}}</ref> <br /><br />

== Open Channel Flow Fundamentals ==

[[File:Open Channel Flow Energy Lines.jpg|thumb|'''Figure 1.''' Conceptual figure used to define terms in the energy equation.<ref>{{cite book|last=Chaudhry|first=M.H.|title=Open-Channel Flow|year=2008|publisher=Springer|___location=New York}}</ref> ]]

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Under steady state flow conditions (e.g. no flood wave), open channel flow can be subdivided into three types of flow: uniform flow, gradually varying flow, and rapidly varying flow. Uniform flow describes a situation where flow depth does not change with distance along the channel. This can only occur in a smooth channel that does not experience any changes in flow, channel geometry, or channel slope. During uniform flow, the flow depth is known as normal depth (yn). This depth is analogous to the terminal velocity of an object in free fall, where gravity and frictional forces are in balance (Moglen, 2013)<ref>{{cite web|last=Moglen|first=G.|title=Lecture Notes from CEE 4324/5894: Open Channel Flow, Virginia Tech|url=http://filebox.vt.edu/users/moglen/ocf/index.html|accessdate=April 24, 2013}}</ref>. Typically, this depth is calculated using the [[Manning formula]]. Gradually varied flow occurs when the change in flow depth per change in flow distance is very small. In this case, hydrostatic relationships developed for uniform flow still apply. Examples of this include the backwater behind an in-stream structure (e.g. dam, sluice gate, weir, etc.), when there is a constriction in the channel, and when there is a minor change in channel slope. Rapidly varied flow occurs when the change in flow depth per change in flow distance is significant. . In this case, hydrostatics relationships are not appropriate for analytical solutions, and continuity of momentum must be employed. Examples of this include large changes in slope like a spillway, abrupt constriction/expansion of flow, or a hydraulic jump.<br /><br />

== Water Surface Profiles (Gradually Varied Flow) ==

Typically, the STM is used to develop “surface water profiles,” or longitudinal representations of channel depth, for channels experiencing gradually varied flow. These transitions can be classified based on reach condition (mild or steep), and also the type of transition being made. Mild reaches occur where normal depth is subcritical (yn > yc) while steep reaches occur where normal depth is supercritical (yn<yc). The transitions are classified by zone. (See figure 3.) <br /><br />

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:* There is a hydrostatic pressure distribution<br /><br />

== Standard Step Method Calculation ==

The STM numerically solves equation 3 through an iterative process. This can be done using the bisection or Newton-Raphson Method, and is essentially solving for total head at a specified ___location using equations 4 and 5 by varying depth at the specified ___location. <ref>{{cite book|last=Chaudhry|first=M.H.|title=Open-Channel Flow|year=2008|publisher=Springer|___location=New York}}</ref>.<br /><br />

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:<math>H_2 = h_{vel} + h_{ele}</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <big>'''Equation 5'''</big><br /><br />

In order to use this technique, it is important to note you must have some understanding of the system you are modeling. For each gradually varied flow transition, you must know both boundary conditions and you must also calculate length of that transition. (e.g. For an M1 Profile, you must find the rise at the downstream boundary condition, the normal depth at the upstream boundary condition, and also the length of the transition.) To find the length of the gradually varied flow transitions, iterate the “step length”, instead of height, at the boundary condition height until equations 4 and 5 agree. (e.g. For an M1 Profile, position 1 would be the downstream condition and you would solve for position two where the height is equal to normal depth.)<br /><br />

=== Newton Raphson Numerical Method ===

[[File:NewtonRaphsonMethod.jpg|NewtonRaphsonMethod]]

=== Conceptual Surface Water Profiles (Sluice Gate) ===

[[File:Sluice Gate Sketch.jpg|thumb| '''Figure 4.''' Illustration of surface water profiles associated with a sluice gate in a mild reach (top) and a steep reach (bottom). ]]

Figure 4 illustrates the different surface water profiles associated with a sluice gate on a mild reach (top) and a steep reach (bottom). Note, the sluice gate induces a choke in the system, causing a “backwater” profile just upstream of the gate. In the mild reach, the [[hydraulic jump]] occurs downstream of the gate, but in the steep reach, the hydraulic jump occurs upstream of the gate. It is important to note that the gradually varied flow equations and associated numerical methods (including the standard step method) cannot accurately model the dynamics of a hydraulic jump <ref>{{cite book|last=Chaudhry|first=M.H.|title=Open-Channel Flow|year=2008|publisher=Springer|___location=New York}}</ref>. See the [[Hydraulic jumps in rectangular channels]] page for more information. Below, an example problem will use conceptual models to build a surface water profile using the STM.

== Example Problem ==

[[File:Problem Statement.jpg|Problem Statement]]<br /><br />

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[[File:Tabularcomparison.jpg|Tabularcomparison]]<br /><br />

== References ==

{{reflist}}

[[:Category:Fluid mechanics]]